Self-force from reconstructed metric perturbations: numerical implementation in Schwarzschild spacetime

TL;DR

This paper introduces a numerical scheme for calculating gravitational self-force in Schwarzschild spacetime using metric reconstruction from the Teukolsky equation, offering improved efficiency and accuracy over previous methods.

Contribution

The paper implements a new method for self-force calculation in Schwarzschild spacetime using metric reconstruction from the Teukolsky equation, demonstrating its accuracy and potential for future Kerr spacetime applications.

Findings

Achieved at least 10^{-7} accuracy with direct numerical integration.

Achieved at least 10^{-9} accuracy with analytical approach.

Methods agree within error margins, validating the approach.

Abstract

We present a first numerical implementation of a new scheme by Pound et al. that enables the calculation of the gravitational self-force in Kerr spacetime from a reconstructed metric-perturbation in a radiation gauge. The numerical task of the metric reconstruction essentially reduces to solving the fully separable Teukolsky equation, rather than having to tackle the linearized Einstein's equations themselves. The method offers significant computational saving compared to existing methods in the Lorenz gauge, and we expect it to become a main workhorse for precision self-force calculations in the future. Here we implement the method for circular orbits on a Schwarzschild background, in order to illustrate its efficacy and accuracy. We use two independent methods for solving the Teukolsky equation, one based on a direct numerical integration, and the other on the analytical approach of Mano, Suzuki, and Takasugi. The relative accuracy of the output self-force is at least $10^{-7}$ using the first method, and at least $10^{-9}$ using the second; the two methods agree to within the error bars of the first. We comment on the relation to a related approach by Shah et al., and discuss foreseeable applications to more generic orbits in Kerr spacetime.